The Chern character is a central construction with incarnations in algebraic topology, representationtheory and algebraic geometry. It is an important tool to probe $K$-theory, which is notoriously hardto compute. In my talk, I will explain, what the categorification of the Chern character is and how wecan use it to show that certain classical constructions in algebraic geometry are of non-commutativeorigin. The category of motives plays the role of $K$-theory in the categorified picture. The categorification

Starting with 't Hooft, physicists have used a ribbon graph expansion to understand certain integrals over spacesof $N \times N$ matrices in the large $N$ limit. This expansion can be deduced from the Feynman diagramexpansion, which relies on the nice structure of moments of a Gaussian measure. We provide a homologicalperspective on this situation: the Batalin-Vilkovisky formalism (which we will outline) provides a homologicalapproach to computing moments, and the Loday-Quillen-Tsygan theorem (which we will explain) gives a

I will talk about a joint work with Si Li on the computation of higher genus B-model for elliptic curves.

I will first formulate the Feynman amplitudes in the higher genus B-model (Kodaira-Spencer theory)in terms of cohomological parings. Then I will discuss properties of the Feynman amplitudes, includingthe origin of their quasi-modularity, the geometric Interpretation of their modular completions, etc.Finally I explain the implication of the cohomological reformation in renormalization.

The M-theory perspective, more specifically the 3d-3d correspondence, leads to interesting predictions for the structure of the Witten-Reshetikhin-Turaev invariants of 3-manifolds and superconformal indices of the 3d SCFTs arising from compactifying five-branes on 3-manifolds. Especially, one expects SL(2,Z) representations to play an important role.

We discuss the recent Hodge--GUE correspondence conjecture on an explicitrelationship between special cubic Hodge integrals over the moduli space of stablealgebraic curves and enumeration of ribbon graphs with even valencies. We sketcha proof of this conjecture based on the Virasoro constraints. We also discuss theconjectural relationship between the cubic Hodge integrals satisfying the localCalabi--Yau condition and the Bogoyavlensky--Toda hierarchy (aka fractionalKdV). The talk is based on a series of joint works with B. Dubrovin, S.-Q. Liu

We will introduce a large class of $\mathcal{N}=1$ superconformal theories, called$S_k$, which is obtained from Gaiotto's $\mathcal{N}=2$ class $S$ via orbifolding. Wewill study the Coulomb branch of the theories in the class byconstructing and analyzing their spectral curves. Using our experiencefrom the $\mathcal{N}=2$ \textsc{agt} correspondence we will search for a 2d/4d relations(\textsc{agt}${}_{k}$) for the $\mathcal{N}=1$ theories of class $S_k$. From the curves we willidentify the 2d \textsc{cft} symmetry algebra and its representations, namely

Nekrasov, Rosly and Shatashvili observed that the generating function of a certain spaceof ${\rm SL}(2)$ opers has a physical interpretation as the effective twisted superpotentialfor a four-dimensional $\mathcal{N}=2$ quantum field theory. In this talk we describe theingredients needed to generalise this observation to higher rank. Important ingredients arespectral networks generated by Strebel differentials and the abelianization method. As anexample we find the twisted superpotential for the $E_6$ Minahan-Nemeschansky theory.

Tropical geometry has been proved successful to study various types of enumerativenumbers, including Gromov-Witten invariants for toric surfaces and Hurwitz numberswith at most two special points. In my talk I will try to give an overview onsome showcase results, recent developments (counting "real'' curves) and relationsto other approaches.

We study two types of actions on King's moduli spaces of quiver representations over afield $k$, and we decompose their fixed loci using group cohomology in order to givemodular interpretations of the components. The first type of action arises by consideringfinite groups of quiver automorphisms. The second is the absolute Galois group of aperfect field $k$ acting on the points of this quiver moduli space valued in an algebraicclosure of $k$; the fixed locus is the set of $k$-rational points, which we decompose

I will discuss a certain class of line defects in four dimensional supersymmetric theorieswith $\mathcal{N}=2$. Many properties of these operators can be rephrased in terms ofquiver representation theory. In particular one can study BPS invariants of a new kind, theso-called framed BPS states, which correspond to bound states of ordinary BPS states withthe defect. Such invariants determine the IR vev of line operators. I will discuss how theseinvariants arise from framed quivers. Time permitting I will also discuss a formalism to study